MIMO different-factor full-form model-free control with parameter self-tuning

ABSTRACT

The invention discloses a MIMO different-factor full-form model-free control method with parameter self-tuning. In view of the limitations of the existing MIMO full-form model-free control method with the same-factor structure, namely, at time k, different control inputs in the control input vector can only use the same values of penalty factor and step-size factors, the invention proposes a MIMO full-form model-free control method with the different-factor structure, namely, at time k, different control inputs in the control input vector can use different values of penalty factors and/or step-size factors, which can solve control problems of strongly nonlinear MIMO systems with different characteristics between control channels widely existing in complex plants. Meanwhile, parameter self-tuning is proposed to effectively address the problem of time-consuming and cost-consuming when tuning the penalty factors and/or step-size factors. Compared with the existing method, the inventive method has higher control accuracy, stronger stability and wider applicability.

FIELD OF THE INVENTION

The present invention relates to the field of automatic control, andmore particularly to MIMO different-factor full-form model-free controlwith parameter self-tuning.

BACKGROUND OF THE INVENTION

In the fields of oil refining, petrochemical, chemical, pharmaceutical,food, paper, water treatment, thermal power, metallurgy, cement, rubber,machinery, and electrical industry, most of the controlled plants, suchas reactors, distillation columns, machines, devices, equipment,production lines, workshops and factories, are essentially MIMO systems(multi-input multi-output systems). Realizing the control of MIMOsystems with high accuracy, strong stability and wide applicability isof great significance to energy saving, consumption reduction, qualityimprovement and efficiency enhancement in industries. However, thecontrol problems of MIMO systems, especially of those with strongnonlinearities, have always been a major challenge in the field ofautomatic control.

MIMO full-form model-free control method is one of the existing controlmethods for MIMO systems. MIMO full-form model-free control method is adata-driven control method, which is used to analyze and design thecontroller depending only on the online measured input data and outputdata instead of any mathematical model information of the MIMOcontrolled plant, and has good application prospects with conciseimplementation, low computational burden and strong robustness. Thetheoretical basis of MIMO full-form model-free control method isproposed by Hou and Jin in Model Free Adaptive Control: Theory andApplications (Science Press, Beijing, China, 2013, p. 116), the controlscheme is given as follows:

${u(k)} = {{u\left( {k - 1} \right)} + \frac{{\Phi_{{Ly} + 1}^{T}(k)}\left( {{\rho_{{Ly} + 1}{e(k)}} - {\sum\limits_{p = 1}^{Ly}{\rho_{p}{\Phi_{p}(k)}\Delta{y\left( {k - p + 1} \right)}}}} \right)}{\lambda + {{\Phi_{{Ly} + 1}(k)}}^{2}} - \frac{{\Phi_{{Ly} + 1}^{T}(k)}{\sum\limits_{p = {{Ly} + 2}}^{{Ly} + {Lu}}{\rho_{p}{\Phi_{p}(k)}\Delta\;{u\left( {k + {Ly} - p + 1} \right)}}}}{\lambda + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$

where u(k) is the control input vector at time k, u(k)=[u₁(k), . . . ,u_(m)(k)]^(T), m is the total number of control inputs (m is a positiveinteger greater than 1), Δu(k)=u(k)−u(k−1); e(k) is the error vector attime k, e(k)=[e₁(k), . . . , e_(n)(k)]^(T), n is the total number ofsystem outputs (n is a positive integer); Δy(k)=y(k)−y(k−1), y(k) is theactual system output vector at time k, y(k)=[y₁(k), . . . ,y_(n)(k)]^(T); Φ(k) is the estimated value of pseudo partitionedJacobian matrix for MIMO system at time k, Φ_(p)(k) is the p-th block ofΦ(k) (p is a positive integer, 1≤p≤Ly+Lu), ∥Φ_(Ly+1)(k)∥ is the 2-normof matrix Φ_(Ly+1)(k); λ is the penalty factor; ρ₁, . . . , ρ_(Ly+Lu)are the step-size factors; Ly is the control output length constant oflinearization and Ly is a positive integer; Lu is the control inputlength constant of linearization and Lu is a positive integer.

The above-mentioned existing MIMO full-form model-free control methodadopts the same-factor structure, namely, at time k, different controlinputs u₁(k), . . . , u_(m)(k) in the control input vector u(k) can onlyuse the same value of penalty factor λ, the same value of step-sizefactor ρ₁, . . . , and the same value of step-size factor ρ_(Ly+Lu).However, when applied to complex plants, such as strongly nonlinear MIMOsystems with different characteristics between control channels, theexisting MIMO full-form model-free control method with the same-factorstructure is difficult to achieve ideal control performance, whichrestricts the popularization and application of MIMO full-formmodel-free control method.

Therefore, in order to break the bottleneck of the existing MIMOfull-form model-free control method with the same-factor structure, thepresent invention proposes a method of MIMO different-factor full-formmodel-free control with parameter self-tuning.

SUMMARY OF THE INVENTION

The present invention addresses the problems cited above, and provides amethod of MIMO different-factor full-form model-free control withparameter self-tuning, the method comprising:

when a controlled plant is a MIMO system, namely a multi-inputmulti-output system, a mathematical formula for calculating the i-thcontrol input u_(i)(k) at time k using said method is as follows:

under the condition that the control input length constant oflinearization Lu>1,

${u_{i}(k)} = {{u_{i}\left( {k - 1} \right)} + \frac{\begin{matrix}{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {{\rho_{i,{{Ly} + 1}}{e_{j}(k)}} - {\sum\limits_{p = 1}^{Ly}\rho_{i,p}}} \right.}} \\\left. \left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta\;{y_{jy}\left( {k - p + 1} \right)}}} \right) \right)\end{matrix}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}} + \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {- {\sum\limits_{p = {{Ly} + 2}}^{Ly}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta\;{u_{iu}\left( {k + {Ly} - p + 1} \right)}}} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$

under the condition that the control input length constant oflinearization Lu=1,

${u_{i}(k)} = {{u_{i}\left( {k - 1} \right)} + \frac{\begin{matrix}{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {{\rho_{i,{{Ly} + 1}}{e_{j}(k)}} - {\sum\limits_{p = 1}^{Ly}\rho_{i,p}}} \right.}} \\\left. \left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta{y_{jy}\left( {k - p + 1} \right)}}} \right) \right)\end{matrix}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$

where k is a positive integer; m is the total number of control inputsin said MIMO system, m is a positive integer greater than 1; n is thetotal number of system outputs in said MIMO system, n is a positiveinteger; i denotes the i-th of the total number of control inputs insaid MIMO system, i is a positive integer, 1≤i≤m; j denotes the j-th ofthe total number of system outputs in said MIMO system, j is a positiveinteger, 1≤j≤n; u_(i)(k) is the i-th control input at time k;Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), iu is a positive integer; e_(j)(k) isthe j-th error at time k, namely the j-th element in the error vectore(k)=[e₁(k), . . . , e_(n)(k)]^(T); Δy_(jy)(k)=y_(jy)(k)−y_(jy)(k−1),y_(jy)(k) is the jy-th actual system output at time k, jy is a positiveinteger; Φ(k) is the estimated value of pseudo partitioned Jacobianmatrix for said MIMO system at time k, Φ_(p)(k) is the p-th block ofΦ(k), ϕ_(j,i,p)(k) is the j-th row and the i-th column of matrixΦ_(p)(k), ∥Φ_(Ly+1)(k)∥ is the 2-norm of matrix Φ_(Ly+1)(k); p is apositive integer, 1≤p≤Ly+Lu; λ₁ is the penalty factor for the i-thcontrol input; ρ_(i,p) is the p-th step-size factor for the i-th controlinput; Ly is the control output length constant of linearization and Lyis a positive integer; Lu is the control input length constant oflinearization and Lu is a positive integer;

for said MIMO system, traversing all values of i in the positive integerinterval [1, m], and calculating the control input vector u(k)=[u₁(k), .. . , u_(m)(k)]^(T) at time k using said method;

said method has a different-factor characteristic; said different-factorcharacteristic is that at least one of the following Ly+Lu+1inequalities holds true for any two unequal positive integers i and x inthe positive integer interval [1, m] during controlling said MIMO systemby using said method:λ_(i)≠λ_(x); ρ_(i,1)≠ρ_(x,1); . . . ; ρ_(i,Ly+Lu)≠ρ_(x,Ly+Lu)

during controlling said MIMO system by using said method, performingparameter self-tuning on the parameters to be tuned in said mathematicalformula for calculating the control input vector u(k)=[u₁(k), . . . ,u_(m)(k)]^(T) at time k; said parameters to be tuned comprise at leastone of: penalty factors λ_(i), and step-size factors ρ_(i,1), . . . ,ρ_(i,Ly+Lu) (i=1, . . . , m).

Said parameter self-tuning adopts neural network to calculate theparameters to be tuned in the mathematical formula of said control inputvector u(k)=[u₁(k), . . . , u_(m)(k)]^(T); when updating the hiddenlayer weight coefficients and output layer weight coefficients of saidneural network, the gradients at time k of said control input vectoru(k)=[u₁(k), . . . , u_(m)(k)]^(T) with respect to the parameters to betuned in their respective mathematical formula are used; the gradientsat time k of u_(i)(k) (i=1, . . . , m) in said control input vectoru(k)=[u₁(k), . . . , u_(m)(k)]^(T) with respect to the parameters to betuned in the mathematical formula of said u_(i)(k) comprise the partialderivatives at time k of u_(i)(k) with respect to the parameters to betuned in the mathematical formula of said u_(i)(k); the partialderivatives at time k of said u_(i)(k) with respect to the parameters tobe tuned in the mathematical formula of said u_(i)(k) are calculated asfollows:

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include penalty factor λ_(i) and the control input lengthconstant of linearization satisfies Lu=1, the partial derivative at timek of u_(i)(k) with respect to said penalty factor λ_(i) is:

$\frac{\partial{u_{i}(k)}}{\partial\lambda_{i}} = \frac{\begin{matrix}{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {{{- \rho_{i,{{Ly} + 1}}}{e_{j}(k)}} + {\sum\limits_{p = 1}^{Ly}\rho_{i,p}}} \right.}} \\\left. \left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta{y_{jy}\left( {k - p + 1} \right)}}} \right) \right)\end{matrix}}{\left( {\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}} \right)^{2}}$

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include penalty factor λ_(i) and the control input lengthconstant of linearization satisfies Lu>1, the partial derivative at timek of u_(i)(k) with respect to said penalty factor λ_(i) is:

$\frac{\partial{u_{i}(k)}}{\partial\lambda_{i}} = {\frac{\begin{matrix}{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {{{- \rho_{i,{{Ly} + 1}}}{e_{j}(k)}} + {\sum\limits_{p = 1}^{Ly}\rho_{i,p}}} \right.}} \\\left. \left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta{y_{jy}\left( {k - p + 1} \right)}}} \right) \right)\end{matrix}}{\left( {\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}} \right)^{2}} + \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {\sum\limits_{p = {{Ly} + 2}}^{{Ly} + {Lu}}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta\;{u_{iu}\left( {k + {Ly} - p + 1} \right)}}} \right)}} \right)}}{\left( {\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}} \right)^{2}}}$

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include step-size factor ρ_(i,p) where 1≤p≤Ly, the partialderivative at time k of u_(i)(k) with respect to said step-size factorρ_(i,p) is:

$\frac{\partial{u_{i}(k)}}{\partial\rho_{i,p}} = {- \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta\;{y_{jy}\left( {k - p + 1} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include Step-size factor ρ_(i,Ly+1), the partial derivative attime k of u_(i)(k) with respect to said step-size factor ρ_(i,Ly+1) is:

$\frac{\partial{u_{i}(k)}}{\partial\rho_{i,{{Ly} + 1}}} = \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}{e_{j}(k)}}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}$

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include step-size factor ρ_(i,p) where Ly+2≤p≤Ly+Lu, and thecontrol input length constant of linearization satisfies Lu>1, thepartial derivative at time k of u_(i)(k) with respect to said step-sizefactor ρ_(i,p) is:

$\frac{\partial{u_{l}(k)}}{\partial\rho_{i,p}} = {- \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta\;{u_{iu}\left( {k + {Ly} - p + 1} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$putting all partial derivatives at time k calculated by said u_(i)(k)with respect to the parameters to be tuned in the mathematical formulaof said u_(i)(k) into the set {gradient of u_(i)(k)}; for said MIMOsystem, traversing all values of i in the positive integer interval [1,m] and obtaining the set {gradient of u₁(k)}, . . . , set {gradient ofu_(m)(k)}, then putting them all into the set {gradient set}; said set{gradient set} is a set comprising all sets {{gradient of u₁(k)}, . . ., {gradient of u_(m)(k)}};

said parameter self-tuning adopts neural network to calculate theparameters to be tuned in the mathematical formula of the control inputvector u(k)=[u₁(k), . . . , u_(m)(k)]^(T); the inputs of said neuralnetwork comprise at least one of: elements in said set {gradient set},and elements in set {error set}; said set {error set} comprises at leastone of: the error vector e(k)=[e₁(k), . . . , e_(n)(k)]^(T), and errorfunction group of element e_(j)(k) (j=1, . . . , n) in said error vectore(k); said error function group of element e_(j)(k) comprises at leastone of: the accumulation of the j-th error at time k and all previoustimes

${\sum\limits_{t = 0}^{k}{e_{j}(t)}},$the first order backward difference of the j-th error e_(j)(k) at time ke_(j)(k)−e_(j)(k−1), the second order backward difference of the j-therror e_(j)(k) at time k e_(j)(k)−2e_(j)(k−1)+e_(j)(k−2), and high orderbackward difference of the j-th error e_(j)(k) at time k.

While adopting the above-mentioned technical scheme, the invention mayadopt or combine the following further technical schemes:

Said j-th error e_(j)(k) at time k is calculated by the j-th errorfunction; independent variables of said j-th error function comprise thej-th desired system output and the j-th actual system output.

Said j-th error function adopts at least one of:e_(j)(k)=y*_(j)(k)−y_(j)(k), e_(j)(k)=y*_(j)(k+1)−y_(j)(k),e_(j)(k)=y_(j)(k)−y*_(j)(k), and e_(j)(k)=y(k)−y*_(j)(k+1), wherey*_(j)(k) is the j-th desired system output at time k, y*_(j)(k+1) isthe j-th desired system output at time k+1, and y_(j)(k) is the j-thactual system output at time k.

Said neural network is BP neural network; said BP neural network adoptsa single hidden layer structure, namely a three-layer network structure,comprising an input layer, a single hidden layer, and an output layer.

Aiming at minimizing a system error function, said neural network adoptsgradient descent method to update the hidden layer weight coefficientsand the output layer weight coefficients, where the gradients arecalculated by system error back propagation; independent variables ofsaid system error function comprise at least one of: elements in theerror vector e(k)=[e₁(k), . . . , e_(n)(k)], n desired system outputs,and n actual system outputs.

Said system error function is defined as

${{\sum\limits_{{jy} = 1}^{n}{a_{jy}{e_{jy}^{2}(k)}}} + {\sum\limits_{{iu} = 1}^{m}{b_{iu}\Delta{u_{iu}^{2}(k)}}}},$where e_(jy)(k) is the jy-th error at time k,Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), u_(iu)(k) is the iu-th control inputat time k, a_(jy) and b_(iu) are two constants greater than or equal tozero, jy and iu are two positive integers.

Said controlled plant comprises at least one of: a reactor, adistillation column, a machine, a device, a set of equipment, aproduction line, a workshop, and a factory.

The hardware platform for running said method comprises at least one of:an industrial control computer, a single chip microcomputer controller,a microprocessor controller, a field programmable gate array controller,a digital signal processing controller, an embedded system controller, aprogrammable logic controller, a distributed control system, a fieldbuscontrol system, an industrial control system based on internet ofthings, and an industrial internet control system.

The inventive MIMO different-factor full-form model-free control methodwith parameter self-tuning uses different penalty factors or step-sizefactors for different control inputs in the control input vector, whichcan solve control problems of strongly nonlinear MIMO systems withdifferent characteristics between control channels widely existing incomplex plants. At the same time, parameter self-tuning is proposed toeffectively address the problem of time-consuming and cost-consumingwhen tuning the penalty factors and/or step-size factors. Compared withthe existing MIMO full-form model-free control method with thesame-factor structure, the inventive MIMO different-factor full-formmodel-free control method with parameter self-tuning has higher controlaccuracy, stronger stability and wider applicability.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic diagram according to the embodiments of theinvention;

FIG. 2 shows a structure diagram of the i-th BP neural network accordingto the embodiments of the invention;

FIG. 3 shows the tracking performance of the first system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the inventive MIMO different-factor full-formmodel-free control method with parameter self-tuning;

FIG. 4 shows the tracking performance of the second system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the inventive MIMO different-factor full-formmodel-free control method with parameter self-tuning;

FIG. 5 shows the control inputs when controlling the two-inputtwo-output MIMO system in the first exemplary embodiment by using theinventive MIMO different-factor full-form model-free control method withparameter self-tuning;

FIG. 6 shows the changes of penalty factor when controlling thetwo-input two-output MIMO system in the first exemplary embodiment byusing the inventive MIMO different-factor full-form model-free controlmethod with parameter self-tuning;

FIG. 7 shows the changes of the first step-size factor when controllingthe two-input two-output MIMO system in the first exemplary embodimentby using the inventive MIMO different-factor full-form model-freecontrol method with parameter self-tuning;

FIG. 8 shows the changes of the second step-size factor when controllingthe two-input two-output MIMO system in the first exemplary embodimentby using the inventive MIMO different-factor full-form model-freecontrol method with parameter self-tuning;

FIG. 9 shows the changes of the third step-size factor when controllingthe two-input two-output MIMO system in the first exemplary embodimentby using the inventive MIMO different-factor full-form model-freecontrol method with parameter self-tuning;

FIG. 10 shows the changes of the fourth step-size factor whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the inventive MIMO different-factor full-formmodel-free control method with parameter self-tuning;

FIG. 11 shows the tracking performance of the first system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the existing MIMO full-form model-free controlmethod with the same-factor structure;

FIG. 12 shows the tracking performance of the second system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the existing MIMO full-form model-free controlmethod with the same-factor structure;

FIG. 13 shows the control inputs when controlling the two-inputtwo-output MIMO system in the first exemplary embodiment by using theexisting MIMO full-form model-free control method with the same-factorstructure;

FIG. 14 shows the tracking performance of the first system output whencontrolling the two-input two-output MIMO system of coal mill in thesecond exemplary embodiment;

FIG. 15 shows the tracking performance of the second system output whencontrolling the two-input two-output MIMO system of coal mill in thesecond exemplary embodiment;

FIG. 16 shows the first control input when controlling the two-inputtwo-output MIMO system of coal mill in the second exemplary embodiment;

FIG. 17 shows the second control input when controlling the two-inputtwo-output MIMO system of coal mill in the second exemplary embodiment;

FIG. 18 shows the changes of penalty factors λ₁ and λ₂ for two controlinputs when controlling the two-input two-output MIMO system of coalmill in the second exemplary embodiment;

FIG. 19 shows the changes of step-size factors ρ_(1,1), ρ_(1,2),ρ_(1,3), ρ_(1,4), ρ_(1,5), ρ_(1,6), ρ_(1,7) for the first control inputwhen controlling the two-input two-output MIMO system of coal mill inthe second exemplary embodiment;

FIG. 20 shows the changes of step-size factors ρ_(2,1), ρ_(2,3),ρ_(2,4), ρ_(2,5), ρ_(2,6), ρ_(2,7) for the second control input whencontrolling the two-input two-output MIMO system of coal mill in thesecond exemplary embodiment.

DETAILED DESCRIPTION OF THE INVENTION

The invention is hereinafter described in detail with reference to theembodiments and accompanying drawings. It is to be understood that otherembodiments may be utilized and structural changes may be made withoutdeparting from the scope of the invention.

FIG. 1 shows a schematic diagram according to the embodiments of theinvention. For a MIMO system with m inputs (m is a positive integergreater than 1) and n outputs (n is a positive integer), the MIMOdifferent-factor full-form model-free control method is adopted tocontrol the system; set the control output length constant oflinearization Ly of the MIMO different-factor full-form model-freecontrol method where Ly is a positive integer; set the control inputlength constant of linearization Lu of the MIMO different-factorfull-form model-free control method where Lu is a positive integer. Forthe i-th control input u_(i)(k) (i=1, . . . , m), the parameters in themathematical formula for calculating u_(i)(k) using the MIMOdifferent-factor full-form model-free adaptive control method includepenalty factor λ_(i) and step-size factors ρ_(i,1), . . . , ρ_(i,Ly+Lu);choose the parameters to be tuned in the mathematical formula ofu_(i)(k), which are part or all of the parameters in the mathematicalformula of u_(i)(k), including at least one of the penalty factor λ_(i),and step-size factors ρ_(i,1), . . . , ρ_(i,Ly+Lu); in the schematicdiagram of FIG. 1, the parameters to be tuned in the mathematicalformula of all control inputs u_(i)(k) (i=1, . . . , m) are the penaltyfactors λ_(i) and step-size factors ρ_(i,1), . . . , ρ_(i,Ly+Lu); theparameters to be tuned in the mathematical formula of u_(i)(k) arecalculated by the i-th BP neural network.

FIG. 2 shows a structure diagram of the i-th BP neural network accordingto the embodiments of the invention; BP neural network can adopt asingle hidden layer structure or a multiple hidden layers structure; forthe sake of simplicity, BP neural network in the diagram of FIG. 2adopts the single hidden layer structure, namely a three-layer networkstructure comprising an input layer, a single hidden layer and an outputlayer; set the number of input layer nodes, hidden layer nodes andoutput layer nodes of the i-th BP neural network; the number of inputlayer nodes of the i-th BP neural network is set to m×(Ly+Lu+1)+n×3,m×(Ly+Lu+1) of which are the elements

$\left\{ {\frac{\partial{u_{iu}\left( {k - 1} \right)}}{\partial\lambda_{iu}},\ \frac{\partial{u_{iu}\left( {k - 1} \right)}}{\partial\rho_{{iu},1}},\ldots\ ,\ \frac{\partial{u_{iu}\left( {k - 1} \right)}}{\partial\rho_{{iu},{{Ly} + {Lu}}}}} \right\}$(iu=1, . . . , m) in set {gradient set}, and the other n×3 are theelements

$\left\{ {{e_{jy}(k)},{\sum\limits_{t = 0}^{k}{e_{jy}(t)}},{{e_{jy}(k)} - {e_{jy}\left( {k - 1} \right)}}} \right\}$(jy=1, . . . , n) in set {error set}; the number of output layer nodesof the i-th BP neural network is no less than the number of parametersto be tuned in the mathematical formula of u_(i)(k); in FIG. 2, thenumber of parameters to be tuned in the mathematical formula of u_(i)(k)is Ly+Lu+1, which are the penalty factor λ_(i) and step-size factorsρ_(i,1), . . . , ρ_(i,Ly+Lu); detailed updating process of hidden layerweight coefficients and output layer weight coefficients of the i-th BPneural network is as follows: in FIG. 2, aiming at minimizing the systemerror function

$\sum\limits_{{jy} = 1}^{n}{e_{jy}^{2}(k)}$with all contributions of n errors comprehensively considered, thegradient descent method is used to update the hidden layer weightcoefficients and the output layer weight coefficients of the i-th BPneural network, where the gradients is calculated by system error backpropagation; in the process of updating the hidden layer weightcoefficients and the output layer weight coefficients of the i-th BPneural network, the elements in set {gradient set}, comprising the set{gradient of u₁(k)}, . . . , set {gradient of u_(m)(k)}, are used,namely the gradients at time

$k\left\{ {\frac{\partial{u_{iu}(k)}}{\partial\lambda_{iu}},\ \frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},1}},\ldots\ ,\ \frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},{{Ly} + {Lu}}}}} \right\}\left( {{{iu} = 1},\ldots\;,m} \right)$of the control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T) withrespect to parameters to be tuned in their respective mathematicalformula.

In combination with the above description of FIG. 1 and FIG. 2, theimplementation steps of the technical scheme in the present inventionare further explained as follows:

mark the current moment as time k; define the difference between thej-th desired system output y*_(j)(k) and the j-th actual system outputy_(j)(k) as the j-th error e_(j)(k); traverse all values of j in thepositive integer interval [1, n] and obtain the error vectore(k)=[e₁(k), . . . , e_(n)(k)], then put them all into the set {errorset}; take the elements

$\left\{ {\frac{\partial{u_{iu}(k)}}{\partial\lambda_{iu}},\frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},1}},\ldots\;,\ \frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},{{Ly} + {Lu}}}}} \right\}\mspace{14mu}\left( {{{iu}\  = 1},\ldots\;,m} \right)$in set {gradient set} and the elements

$\left\{ {{e_{jy}(k)},{\sum\limits_{t = 0}^{k}{e_{jy}(t)}},{{e_{jy}(k)} - {e_{jy}\left( {k - 1} \right)}}} \right\}\mspace{14mu}\left( {{{jy} = 1},\ldots\;,n} \right)$in set {error set} as the inputs of the i-th BP neural network; obtainthe parameters to be tuned in the mathematical formula for calculatingu_(i)(k) using the MIMO different-factor full-form model-free controlmethod by the output layer of the i-th BP neural network through forwardpropagation; based on the error vector e(k) and the parameters to betuned in the mathematical formula of u_(i)(k), calculate the i-thcontrol input u_(i)(k) at time k using the MIMO different-factorfull-form model-free adaptive control method; traverse all values of iin the positive integer interval [1, m] and obtain the control inputvector u(k) [u₁(k), . . . , u_(m)(k)]^(T) at time k; for u_(i)(k) in thecontrol input vector u(k), calculate all partial derivatives of u_(i)(k)with respect to the parameters to be tuned in the mathematical formula,and put them all into the set {gradient of u_(i)(k)}; traverse allvalues of i in the positive integer interval [1, m] and obtain the set{gradient of u₁(k)}, . . . , set {gradient of u_(m)(k)}, and put themall into the set {gradient set}; then, aiming at minimizing the systemerror function

$\sum\limits_{{jy} = 1}^{n}{e_{jy}^{2}(k)}$with all contributions of n errors comprehensively considered and usingthe gradients

$\left\{ {\frac{\partial{u_{iu}(k)}}{\partial\lambda_{iu}},\frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},1}},\ldots\;,\ \frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},{{Ly} + {Lu}}}}} \right\}\mspace{14mu}\left( {{{iu}\  = 1},\ldots\;,m} \right)$in set {gradient set}, update the hidden layer weight coefficients andthe output layer weight coefficients of the i-th BP neural network usingthe gradient descent method, where the gradients is calculated by systemerror back propagation; traverse all values of i in the positive integerinterval [1, m] and update the hidden layer weight coefficients and theoutput layer weight coefficients of all m BP neural networks; obtain then actual system outputs at next time by applying the control inputvector u(k) into the controlled plant, and then repeat the stepsdescribed in this paragraph for controlling the MIMO system at nextsampling time.

Two exemplary embodiments of the invention are given for furtherexplanation.

The First Exemplary Embodiment

A two-input two-output MIMO system, which has the complexcharacteristics of non-minimum phase nonlinear system, is adopted as thecontrolled plant, and it belongs to the MIMO system that is particularlydifficult to control:

${y_{1}(k)} = {\frac{{2.5}{y_{1}\left( {k - 1} \right)}{y_{1}\left( {k - 2} \right)}}{1 + {y_{1}^{2}\left( {k - 1} \right)} + {y_{2}^{2}\left( {k - 2} \right)} + {y_{1}^{2}\left( {k - 3} \right)}} + {{0.0}9{u_{1}\left( {k - 1} \right)}{u_{1}\left( {k - 2} \right)}} + {1.2{u_{1}\left( {k - 1} \right)}} + {1.6{u_{1}\left( {k - 3} \right)}} + {{0.5}{u_{2}\left( {k - 1} \right)}} + {0.7{\sin\left( {{0.5}\left( {{y_{1}\left( {k - 1} \right)} + {y_{1}\left( {k - 2} \right)}} \right)} \right)}{\cos\left( {{0.5}\left( {{y_{1}\left( {k - 1} \right)} + {y_{1}\left( {k - 2} \right)}} \right)} \right)}}}$${y_{2}(k)} = {\frac{5{y_{2}\left( {k - 1} \right)}{y_{2}\left( {k - 2} \right)}}{1 + {y_{2}^{2}\left( {k - 1} \right)} + {y_{1}^{2}\left( {k - 2} \right)} + {y_{2}^{2}\left( {k - 3} \right)}} + {u_{2}\left( {k - 1} \right)} + {{1.1}{u_{2}\left( {k - 2} \right)}} + {1.4{u_{2}\left( {k - 3} \right)}} + {{0.5}{u_{1}\left( {k - 1} \right)}}}$

The desired system outputs y*(k) are as follows:y* ₁(k)=5 sin(k/50)+2 cos(k/20)y* ₂(k)=2 sin(k/50)+5 cos(k/20)

In this embodiment, m=n=2.

The control output length constant of linearization Ly is usually setaccording to the complexity of the controlled plant and the actualcontrol performance, generally between 1 and 5, while large Ly will leadto massive calculation, so it is usually set to 1 or 3; in thisembodiment, Ly=1; the control input length constant of linearization Luis usually set according to the complexity of the controlled plant andthe actual control performance, generally between 1 and 10, while smallLu will affect the control performance and large Lu will lead to massivecalculation, so it is usually set to 3 or 5; in this embodiment, Lu=3.

In view of the above exemplary embodiment, five experiments are carriedout for comparison and verification. In order to compare the controlperformance of the five experiments clearly, root mean square error(RMSE) is adopted as the control performance index for evaluation:

${{RMSE}\left( e_{j} \right)} = \sqrt{\frac{1}{N}{\sum\limits_{k = 1}^{N}{e_{j}^{2}(k)}}}$

where e_(j)(k)=y*_(j)(k)−y_(j)(k), y*_(j)(k) is the j-th desired systemoutput at time k, y_(j)(k) is the j-th actual system output at time k.The smaller the value of RMSE(e_(j)) is, the smaller the error betweenthe j-th actual system output and the j-th desired system output is, andthe better the control performance gets.

The hardware platform for running the inventive control method is theindustrial control computer.

The first experiment (RUN1): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to16, 10 of which are the elements

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,4}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,3}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,4}}} \right\}$in set {gradient set}, and the other 6 are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 6; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is both set to 5, where the first BP neuralnetwork outputs penalty factor λ₁ and step-size factors ρ_(1,1),ρ_(1,2), ρ_(1,3), ρ_(1,4), and the second BP neural network outputspenalty factor λ₂ and step-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3),ρ_(2,4); the inventive MIMO different-factor full-form model-freecontrol method with parameter self-tuning is adopted to control theabove two-input two-output MIMO system; the tracking performance of thefirst system output and second system output are shown in FIG. 3 andFIG. 4, respectively, and the control inputs are shown in FIG. 5; FIG. 6shows the changes of penalty factor, FIG. 7 shows the changes of thefirst step-size factor, FIG. 8 shows the changes of the second step-sizefactor, FIG. 9 shows the changes of the third step-size factor and FIG.10 shows the changes of the fourth step-size factor; evaluate thecontrol method from the control performance indexes: the RMSE(e₁) of thefirst system output in FIG. 3 is 0.5243, and the RMSE(e₂) of the secondsystem output in FIG. 4 is 0.8310; evaluate the control method from thedifferent-factor characteristic: the changes of penalty factor in FIG. 6basically do not overlap, indicating that the different-factorcharacteristic for penalty factor is significant when controlling theabove two-input two-output MIMO system, and the changes of step-sizefactors in FIGS. 7, 8, 9, 10 basically do not overlap, indicating thatthe different-factor characteristic for step-size factors aresignificant when controlling the above two-input two-output MIMO system.

The second experiment (RUN2): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to10, all of which are the elements

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,4}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,3}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,4}}} \right\}$in set {gradient set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 6; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is both set to 5, where the first BP neuralnetwork outputs penalty factor λ₁ and step-size factors ρ_(1,1),ρ_(1,2), ρ_(1,3), ρ_(1,4), and the second BP neural network outputspenalty factor λ₂ and step-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3),ρ_(2,4); the inventive MIMO different-factor full-form model-freecontrol method with parameter self-tuning is adopted to control theabove two-input two-output MIMO system; evaluate the control method fromthe control performance indexes: the RMSE(e₁) of the first system outputis 0.5409, and the RMSE(e₂) of the second system output is 0.9182.

The third experiment (RUN3): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to6, all of which are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 6; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is set to 5, where the first BP neural networkoutputs penalty factor λ₁ and step-size factors, ρ_(1,1), ρ_(1,2),ρ_(1,3), ρ_(1,4), and the second BP neural network outputs penaltyfactor λ₂ and step-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3), ρ_(2,4); theinventive MIMO different-factor full-form model-free control method withparameter self-tuning is adopted to control the above two-inputtwo-output MIMO system; evaluate the control method from the controlperformance indexes: the RMSE(e₁) of the first system output is 0.5412,and the RMSE(e₂) of the second system output is 0.9376.

The fourth experiment (RUN4): the penalty factors λ₁, λ₂ and step-sizefactors ρ_(1,1), ρ_(1,2), ρ_(1,3), ρ_(1,4) are fixed, and only thestep-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3), ρ_(2,4) for the secondcontrol input are chosen for the parameters to be tuned, therefore, onlyone BP neural network is adopted here; the number of input layer nodesof the BP neural network is set to 6, all of which are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the BP neuralnetwork is set to 6; the number of output layer nodes of the BP neuralnetwork is set to 4, where the outputs are step-size factors ρ_(2,1),ρ_(2,2), ρ_(2,3), ρ_(2,4); the inventive MIMO different-factor full-formmodel-free control method with parameter self-tuning is adopted tocontrol the above two-input two-output MIMO system; evaluate the controlmethod from the control performance indexes: the RMSE(e₁) of the firstsystem output is 0.5421, and the RMSE(e₂) of the second system output is0.9850.

The fifth experiment (RUN5): the existing MIMO full-form model-freecontrol method is adopted control the above two-input two-output MIMOsystem; set the penalty factor λ=1.00, and the step-size factorsρ₁=ρ₂=ρ₃=ρ₄=0.50; the tracking performance of the first system outputand the second system output are shown in FIG. 11 and FIG. 12,respectively, and the control inputs are shown in FIG. 13; evaluate thecontrol method from the control performance indexes: the RMSE(e₁) of thefirst system output in FIG. 11 is 0.5433, and the RMSE(e₂) of the secondsystem output in FIG. 12 is 1.0194.

The comparison results of control performance indexes of the fiveexperiments are shown in Table 1; the results of the first experiment tothe fourth experiment (RUN1, RUN2, RUN3, RUN4) using the inventivecontrol method are superior to those of the fifth experiment (RUN5)using the existing MIMO full-form model-free control method with thesame-factor structure; by comparing FIG. 4 and FIG. 12, it can be seenthat the tracking performance of the second system output controlled bythe inventive controller has a significant improvement, indicating thatthe inventive MIMO different-factor full-form model-free control methodwith parameter self-tuning has higher control accuracy, strongerstability and wider applicability.

TABLE 1 Comparison Results of The Control Performance The first systemoutput The second system output RMSE(e₁) Improvement RMSE(e₂)Improvement RUN1 0.5243 3.497% 0.8310 18.484% RUN2 0.5409 0.442% 0.91829.927% RUN3 0.5412 0.387% 0.9376 8.024% RUN4 0.5421 0.221% 0.9850 3.375%RUN5 0.5433 Baseline 1.0194 Baseline

The Second Exemplary Embodiment

A coal mill is a very important set of equipment that pulverizes rawcoal into fine powder, providing fine powder for the pulverized coalfurnace. Realizing the control of coal mill with high accuracy, strongstability and wide applicability is of great significance to ensure thesafe and stable operation of thermal power plant.

The two-input two-output MIMO system of coal mill, which has the complexcharacteristics of nonlinearity, strong coupling and time-varying, isadopted as the controlled plant, and it belongs to the MIMO system thatis particularly difficult to control. Two control inputs u₁(k) and u₂(k)of the coal mill are hot air flow (controlled by the opening of hot airgate) and recycling air flow (controlled by the opening of recycling airgate), respectively. Two system outputs y₁(k) and y₂(k) of the coal millare outlet temperature (° C.) and inlet negative pressure (Pa),respectively. The initial conditions of the coal mill are: u₁(0)=80%,u₂(0)=40%, y₁(0)=70° C., y₂ (0)=−400 Pa. At the 50th second, in order tomeet the needs of on-site conditions adjustment in thermal power plant,the desired system output y*₁(50) is adjusted from 70° C. to 80° C., andthe desired system output y*₂(k) is required to remain unchanged at −400Pa. In view of the above typical conditions in the thermal power plant,two experiments are carried out for comparison and verification. In thisembodiment, m=n=2, the control output length constant of linearizationLy is set to 3, and the control input length constant of linearizationLu is set to 4. The hardware platform for running the inventive controlmethod is the industrial control computer.

The sixth experiment (RUN6): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to22, 16 of which are the elements

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,4}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,5}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,6}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,7}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,3}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,4}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,5}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,6}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,7}}} \right\}$in set {gradient set}, and the other 6 are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 20; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is both set to 8, where the first BP neuralnetwork outputs penalty factor λ₁, and step-size factors ρ_(1,1),ρ_(1,2), ρ_(1,3), ρ_(1,4), ρ_(1,5), ρ_(1,6), ρ_(1,7), and the second BPneural network outputs penalty factor λ₂ and step-size factors ρ_(2,1),ρ_(2,2), ρ_(2,3), ρ_(2,4), ρ_(2,5), ρ_(2,6), ρ_(2,7); the inventive MIMOdifferent-factor full-form model-free control method with parameterself-tuning is adopted to control the above two-input two-output MIMOsystem; the tracking performance of the first output is shown as RUN6 inFIG. 14, the tracking performance of the second output is shown as RUN6in FIG. 15, the first control input is shown as RUN6 in FIG. 16, and thesecond control input is shown as RUN6 in FIG. 17; FIG. 18 shows thechanges of penalty factors λ_(i), and λ₂ for two control inputs, FIG. 19shows the changes of step-size factors ρ_(1,1), ρ_(1,2), ρ_(1,3),ρ_(1,4), ρ_(1,5), ρ_(1,6), ρ_(1,7) for the first control input, and FIG.20 shows the changes of step-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3),ρ_(2,4), ρ_(2,5), ρ_(2,6), ρ_(2,7) for the second control input;evaluate the control method from the control performance indexes: theRMSE(e₁) of the first system output, RUN6 in FIG. 14, is 1.8903, and theRMSE(e₂) of the second system output, RUN6 in FIG. 15, is 0.0791;evaluate the control method from the different-factor characteristic:the changes of penalty factors for two control inputs in FIG. 18 do notoverlap at all, indicating that the different-factor characteristic forpenalty factor is significant when controlling the above two-inputtwo-output MIMO system, and the changes of step-size factors for twocontrol inputs in FIG. 19 and FIG. 20 basically do not overlap,indicating that the different-factor characteristic for step-sizefactors are significant when controlling the above two-input two-outputMIMO system.

The seventh experiment (RUN7): the MIMO different-factor full-formmodel-free control method with fixed parameters is adopted to controlthe above two-input two-output MIMO system; set the parameters value forcalculating the first control input: the penalty factor λ₁=0.03, thestep-size factors ρ_(1,1)=2.05, ρ_(1,2)=1.04, ρ_(1,3)=1.00,ρ_(1,4)=1.03, ρ_(1,5)=0.88, ρ_(1,6)=1.07, ρ_(1,7)=0.88; set theparameters value for calculating the second control input: the penaltyfactor λ₂=0.02, the step-size factors ρ_(2,1)=1.89, ρ_(2,2)=0.96,ρ_(2,3)=0.92, ρ_(2,4)=0.96, ρ_(2,5)=0.80, ρ_(2,6)=0.99, ρ_(2,7)=0.80;the tracking performance of the first system output is shown as RUN7 inFIG. 14, the tracking performance of the second system output is shownas RUN7 in FIG. 15, the first control input is shown as RUN7 in FIG. 16,and the second control input is shown as RUN7 in FIG. 17; evaluate thecontrol method from the control performance indexes: the RMSE(e₁) of thefirst system output, RUN7 in FIG. 14, is 2.0636, and the RMSE(e₂) of thesecond system output, RUN7 in FIG. 15, is 0.0903.

The eighth experiment (RUN8): the existing MIMO full-form model-freecontrol method with the same-factor structure is adopted to control theabove two-input two-output MIMO system; set the penalty factor λ=0.03,the step-size factors ρ₁=2, ρ₂=ρ₃=ρ₄=ρ₅=ρ₆=ρ₇=1; the trackingperformance of the first system output is RUN8 in FIG. 14, the trackingperformance of the second system output is RUN8 in FIG. 15, the firstcontrol input is the RUN8 in FIG. 16, and the second control input isRUN8 in FIG. 17; evaluate the control method from the controlperformance indexes: the RMSE(e₁) of the first system output, RUN8 inFIG. 14, is 2.1678, and the RMSE(e₂) of the second system output, RUN8in FIG. 15, is 0.1001.

The comparison results of control performance indexes of the threeexperiments are shown in Table 2; the results of the sixth experiment(RUN6) using the inventive control method are superior to those of theseventh experiment (RUN7) using the MIMO different-factor full-formmodel-free control method with fixed parameters, and are moresignificantly superior to those of the eighth experiment (RUN8) usingthe existing MIMO full-form model-free control method with thesame-factor structure, which have a significant improvement, indicatingthat the inventive MIMO different-factor full-form model-free controlmethod with parameter self-tuning has higher control accuracy, strongerstability and wider applicability.

TABLE 2 Comparison Results of The Control Performance of Coal Mill Thefirst system output The second system output RMSE(e₁) ImprovementRMSE(e₂) Improvement RUN6 1.8903 12.801% 0.0791 20.979% RUN7 2.06364.807% 0.0903 9.790% RUN8 2.1678 Baseline 0.1001 Baseline

Furthermore, the following six points should be noted in particular:

(1) In the fields of oil refining, petrochemical, chemical,pharmaceutical, food, paper, water treatment, thermal power, metallurgy,cement, rubber, machinery, and electrical industry, most of thecontrolled plants, such as reactors, distillation columns, machines,equipment, devices, production lines, workshops and factories, areessentially MIMO systems; some of these MIMO systems have the complexcharacteristics of non-minimum phase nonlinear system, which belong tothe MIMO systems that are particularly difficult to control; forexample, the continuous stirred tank reactor (CSTR), commonly used inoil refining, petrochemical, chemical, etc. is a two-input two-outputMIMO system, where the two inputs are feed flow and cooling water flow,and the two outputs are product concentration and reaction temperature;when the chemical reaction has strong exothermic effect, the continuousstirred tank reactor (CSTR) is a MIMO system with complexcharacteristics of non-minimum phase nonlinear system, which isparticularly difficult to control. In the first exemplary embodiment,the controlled plant with two inputs and two outputs also has thecomplex characteristic of non-minimum phase nonlinear system and belongsto the MIMO system that is particularly difficult to control; theinventive controller is capable of controlling the plant with highaccuracy, strong stability and wide applicability, indicating that itcan also achieve high accuracy, strong stability and wide applicabilitycontrol on complex MIMO systems such as reactors, distillation columns,machines, equipment, devices, production lines, workshops, factories,etc.

(2) In the first and second exemplary embodiments, the hardware platformfor running the inventive controller is the industrial control computer;in practical applications, according to the specific circumstance, asingle chip microcomputer controller, a microprocessor controller, afield programmable gate array controller, a digital signal processingcontroller, an embedded system controller, a programmable logiccontroller, a distributed control system, a fieldbus control system, anindustrial control system based on internet of things, or an industrialinternet control system, can also be used as the hardware platform forrunning the inventive control method.

(3) In the first and second exemplary embodiments, the j-th errore_(j)(k) is defined as the difference between the j-th desired systemoutput y*_(j)(k) and the j-th actual system output y_(j)(k), namelye_(j)(k)=y*_(j)(k)−y_(j)(k), which is only one of the methods forcalculating the j-th error; the j-th error e_(j)(k) can also be definedas the difference between the j-th desired system output y*_(j)(k+1) attime k+1 and the j-th actual system output y_(j)(k), namelye_(j)(k)=y*_(j)(k+1)−y_(j)(k); the j-th error e_(j)(k) can also bedefined by other methods whose independent variables include the j-thdesired system output and the j-th actual system output, for example,

${{e_{j}(k)} = {\frac{{y_{j}^{*}\left( {k + 1} \right)} + {y_{j}^{*}(k)}}{2} - {y_{j}(k)}}};$for the controlled plants in the first and second exemplary embodiments,all different definitions of the error function can achieve good controlperformance.

(4) The inputs of BP neural network include at least one of: theelements in set {gradient set}, and the elements in set {error set};when the inputs of BP neural network include the elements in set{gradient set}, the gradients at time k−1 are used in the firstexemplary embodiment, namely

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,4}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,3}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,4}}} \right\};$in practical applications, the gradients at more time can be furtheradded according to the specific situation; for example, the gradients attime k−2 can be added, namely

$\left\{ {\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\rho_{1,4}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\rho_{2,3}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\rho_{2,4}}} \right\};$when the inputs of BP neural network include the elements in set {errorset}, the error function group

$\left\{ {{e_{1}(k)},{\sum\limits_{t - 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t - 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$is used in the first and second exemplary embodiments; in practicalapplications, more error function groups can be further added to the set{error set} according to the specific situation; for example, the secondorder backward difference of the j-th error e_(jy)(k), namely{e₁(k)−2e₁(k−1)+e₁(k−2), e₂(k)−2e₂(k−1)+e₂(k−2)}, can also be added intothe inputs of BP neural network; furthermore, the inputs of BP neuralnetwork include, but is not limited to, the elements in set {gradientset} and set {error set}; for example, {u₁(k−1), u₂(k−1)} can also beadded into the inputs of BP neural network; for the controlled plants inthe first and second exemplary embodiments, the inventive controller canachieve good control performance with the increasing of the number ofinput layer nodes of BP neural network, and in most cases it canslightly improve the control performance, but at the same time itincreases the computational burden; therefore, the number of input layernodes of BP neural network can be set to a reasonable number accordingto specific conditions in practical applications.

(5) In the first and second exemplary embodiments, when updating thehidden layer weight coefficients and the output layer weightcoefficients with the objective of minimizing the system error function,all contributions of n errors are comprehensively considered in saidsystem error function

${\sum\limits_{{jy} = 1}^{n}{e_{jy}^{2}(k)}},$which is just one of the system error functions; said system errorfunction can also adopt other functions whose independent variablesinclude any one or any combination of the elements in n errors, ndesired system outputs and n actual system outputs; for example, saidsystem error function can adopt another way of

${\sum\limits_{{jy} = 1}^{n}{e_{jy}^{2}(k)}},$such as

${\sum\limits_{{jy} = 1}^{n}{\left( {{y_{jy}^{*}(k)} - {y_{jy}(k)}} \right)^{2}\mspace{14mu}{or}\mspace{14mu}{\sum\limits_{{jy} = 1}^{n}\left( {{y_{jy}^{*}\left( {k + 1} \right)} - {y_{jy}(k)}} \right)^{2}}}};$for another example, said system error function can adopt

${{\sum\limits_{{jy} = 1}^{n}{a_{jy}{e_{jy}^{2}(k)}}} + {\sum\limits_{{iu} = 1}^{m}{b_{iu}\Delta\;{u_{iu}^{2}(k)}}}},$where e_(jy)(k) is the jy-th error at time k,Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), u_(iu)(k) is the iu-th control inputat time k, a_(jy) and b_(iu) are two constants greater than or equal to0, jy and iu are two positive integers; obviously, when b_(iu) equals tozero, said system error function only considers the contribution ofe_(jy) ²(k), indicating that the objective is to minimize the systemerror and pursue high control accuracy; when b_(iu) is greater thanzero, said system error function considers the contributions of e_(jy)²(k) and Δu_(iu) ²(k) simultaneously, indicating that the objective isnot only to minimize the system error but also to minimize the varianceof control inputs, that is, to pursue high control accuracy and stablecontrol; for the controlled plants in the first and second exemplaryembodiments, all different system error functions can achieve goodcontrol performance; compared with the system error function onlyconsidering the contribution of e_(jy) ²(k), the control accuracy isslightly reduced while the handling stability is improved when thecontributions of e_(jy) ²(k) and Δu_(iu) ²(k) are taken into accountsimultaneously in the system error function.

(6) The parameters to be tuned in said MIMO different-factor full-formmodel-free control method with parameter self-tuning include at leastone of: penalty factors λ_(i), and step-size factors ρ_(i,1), . . . ,ρ_(i,Ly+LU) (i=1, . . . , m); in the first exemplary embodiment, allpenalty factors λ₁, λ₂ and step-size factors ρ_(1,1), ρ_(1,2), ρ_(1,3),ρ_(1,4), ρ_(2,1), ρ_(2,2), ρ_(2,3), ρ_(2,4) are self-tuned in the firstexperiment to the third experiment; in the fourth experiment, only thestep-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3), ρ_(2,4) for the secondcontrol input are self-tuned, while the penalty factors λ₁, λ₂ andstep-size factors ρ_(1,1), ρ_(1,2), ρ_(1,3), ρ_(1,4) are fixed; inpractical applications, any combination of the parameters to be tunedcan be chosen according to the specific situation; in addition, saidparameters to be tuned include, but are not limited to: penalty factorsλ_(i), and step-size factors ρ_(i,1), . . . , ρ_(i,Ly+LU) (i=1, . . . ,m); for example, said parameters to be tuned can also include theparameters for calculating the estimated value of pseudo partitionedJacobian matrix Φ(k) for said MIMO system.

It should be appreciated that the foregoing is only preferredembodiments of the invention and is not for use in limiting theinvention. Any modification, equivalent substitution, and improvementwithout departing from the spirit and principle of this invention shouldbe covered in the protection scope of the invention.

The invention claimed is:
 1. A method of MIMO different-factor full-form model-free control with parameter self-tuning, executed on a hardware platform for controlling a controlled plant being a multi-input multi-output (MIMO) system, wherein the MIMO system having a predetermined number of control inputs and a predetermined number of system outputs, said controlled plant comprises at least one of: a reactor, a distillation column, a machine, a device, a set of equipment, a production line, a workshop, and a factory, said hardware platform comprises at least one of: an industrial control computer, a single chip microcomputer controller, a microprocessor controller, a field programmable gate array controller, a digital signal processing controller, an embedded system controller, a programmable logic controller, a distributed control system, a fieldbus control system, an industrial control system based on internet of things, and an industrial internet control system, said method comprising: calculating the i-th control input u₁(k) at time k as follows: under the condition that the control input length constant of linearization Lu>1, ${u_{i}(k)} = {{u_{i}\left( {k - 1} \right)} + \frac{\begin{matrix} {\sum\limits_{j = 1}^{n}{\phi_{j,i,{{Ly} + 1}}(k)}} \\ \left( {{\rho_{i,{{Ly} + 1}}{e_{j}(k)}} - {\sum\limits_{p = 1}^{Ly}{\rho_{i,p}\left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta{y_{jy}\left( {k - p + 1} \right)}}} \right)}}} \right) \end{matrix}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}} + \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {- {\sum\limits_{p - {Ly} + 2}^{{Ly} + {Lu}}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k + {Ly} - p + 1} \right)}}} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$ under the condition that the control input length constant of linearization Lu=1, ${u_{i}(k)} = {{u_{i}\left( {k - 1} \right)} + \frac{\begin{matrix} {\sum\limits_{j = 1}^{n}{\phi_{j,i,{{Ly} + 1}}(k)}} \\ \left( {{\rho_{i,{{Ly} + 1}}{e_{j}(k)}} - {\sum\limits_{p = 1}^{L_{y}}{\rho_{i,p}\left( {\sum\limits_{{JY} - 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta{y_{jy}\left( {k - p + 1} \right)}}} \right)}}} \right) \end{matrix}}{\lambda_{i} + {{\Phi_{{L\gamma} + 1}(k)}}^{2}}}$ where k is a positive integer; m is the total number of control inputs in said MIMO system, m is a positive integer greater than 1; n is the total number of system outputs in said MIMO system, n is a positive integer; i denotes the i-th of the total number of control inputs in said MIMO system, i is a positive integer, 1≤i≤m; j denotes the j-th of the total number of system outputs in said MIMO system, j is a positive integer, 1≤j≤n; u_(i)(k) is the i-th control input at time k; Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), iu is a positive integer; e_(j)(k) is the j-th error at time k, namely the j-th element in the error vector e(k)=[e₁(k), . . . , e_(n)(k)]^(T); Δy_(jy)(k)=y_(jy)(k)−y_(jy)(k−1), y_(jy)(k) is the jy-th actual system output at time k, jy is a positive integer; Φ(k) is the estimated value of pseudo partitioned Jacobian matrix for said MIMO system at time k, Φ_(p)(k) is the p-th block of Φ(k), ϕ_(j,i,p)(k) is the j-th row and the i-th column of matrix Φ_(p)(k), ∥Φ_(Ly+1)(k)∥ is the 2-norm of matrix Φ_(Ly+1)(k); p is a positive integer, 1≤p≤Ly+Lu; λ₁ is the penalty factor for the i-th control input; ρ_(i,p) is the p-th step-size factor for the i-th control input; Ly is the control output length constant of linearization and Ly is a positive integer; Lu is the control input length constant of linearization and Lu is a positive integer; for said MIMO system, calculating the control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T) by traversing all values of i in the positive integer interval [1, m]; said method has a different-factor characteristic; said different-factor characteristic is that at least one of the following Ly+Lu+1 inequalities holds true for any two unequal positive integers i and x in the positive integer interval [1, m] during controlling said MIMO system by using said method: λ_(i)≠λ_(x); ρ_(i,1)≠ρ_(x,1); . . . ; ρ_(i,Ly+Lu)≠ρ_(x,Ly+Lu); during controlling said MIMO system by using said method, performing parameter self-tuning on the parameters to be tuned in the control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T) at time k; said parameters to be tuned comprise at least one of: penalty factors λ_(i), and step-size factors ρ_(i,1), . . . , ρ_(i,Ly+Lu) (i=1, . . . , m); and obtaining the system outputs from the MIMO system by adjusting the control inputs of the MIMO system based on the calculated control input vector, such that the system outputs of the MIMO system approach desired system outputs to be received by the hardware platform.
 2. The method as claimed in claim 1 wherein said parameter self-tuning adopts neural network to calculate the parameters to be tuned in the mathematical formula of said control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T); when updating the hidden layer weight coefficients and output layer weight coefficients of said neural network, the gradients at time k of said control input vector u(k)=[u₁(k), . . . , u_(m) (k)]^(T) with respect to the parameters to be tuned in their respective mathematical formula are used; the gradients at time k of u_(i)(k) (i=1, . . . , m) in said control input vector u(k)=[u₁(k), . . . , u_(m) (k)]^(T) with respect to the parameters to be tuned in the mathematical formula of said u_(i)(k) comprise the partial derivatives at time k of u_(i)(k) with respect to the parameters to be tuned in the mathematical formula of said u_(i)(k); the partial derivatives at time k of said u_(i)(k) with respect to the parameters to be tuned in the mathematical formula of said u_(i)(k) are calculated as follows: when the parameters to be tuned in the mathematical formula of said u_(i)(k) include penalty factor λ_(i), and the control input length constant of linearization satisfies Lu=1, the partial derivative at time k of u_(i)(k) with respect to said penalty factor λ_(i) is: $\frac{\partial{u_{i}(k)}}{\partial\lambda_{i}} = \frac{\begin{matrix} {\sum\limits_{j = 1}^{n}{\phi_{j,i,{{Ly} + 1}}(k)}} \\ \left( {{{- \rho_{i,{{Ly} + 1}}}{e_{j}(k)}} + {\sum\limits_{p = 1}^{Ly}{\rho_{i,p}\left( {\sum\limits_{{jy} - 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta{y_{jy}\left( {k - p + 1} \right)}}} \right)}}} \right) \end{matrix}}{\left( {\lambda_{i} + {{\Phi_{{L\gamma} + 1}(k)}}^{2}} \right)^{2}}$ when the parameters to be tuned in the mathematical formula of said u_(i)(k) include penalty factor λ_(i), and the control input length constant of linearization satisfies Lu>1, the partial derivative at time k of u_(i)(k) with respect to said penalty factor λ_(i) is: $\frac{\partial{u_{i}(k)}}{\partial\lambda_{i}} = {\frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {{{- \rho_{i,{{Ly} + 1}}}{e_{j}(k)}} + {\sum\limits_{p = 1}^{Ly}{\rho_{i,p}\left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta\;{y_{jy}\left( {k - p + 1} \right)}}} \right)}}} \right)}}{\left( {\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}} \right)^{2}} + \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {\sum\limits_{p = {{Ly} + 2}}^{{Ly} + {Lu}}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta\;{u_{iu}\left( {k + {Ly} - p + 1} \right)}}} \right)}} \right)}}{\left( {\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}} \right)^{2}}}$ when the parameters to be tuned in the mathematical formula of said u_(i)(k) include step-size factor ρ_(i,p) where 1≤p≤Ly, the partial derivative at time k of u_(i)(k) with respect to said step-size factor ρ_(i,p) is: $\frac{\partial{u_{i}(k)}}{\partial\rho_{i,p}} = {- \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {\sum\limits_{{jy} = 1}^{n}{{\phi_{j,{jy},p}(k)}\Delta{y_{jy}\left( {k - p + 1} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$ when the parameters to be tuned in the mathematical formula of said u_(i)(k) include step-size factor ρ_(i,Ly+1), the partial derivative at time k of u_(i)(k) with respect to said step-size factor ρ_(i,Ly+1) is: $\frac{\partial{u_{i}(k)}}{\partial\rho_{i,{{Ly} + 1}}} = \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}{e_{j}(k)}}}{\lambda_{i} + {{\Phi_{{L\gamma} + 1}(k)}}^{2}}$ when the parameters to be tuned in the mathematical formula of said u_(i)(k) include step-size factor ρ_(i,p) where Ly+2≤p≤Ly+Lu, and the control input length constant of linearization satisfies Lu>1, the partial derivative at time k of u_(i)(k) with respect to said step-size factor ρ_(i,p) is: $\frac{\partial{u_{i}(k)}}{\partial\rho_{i,p}} = {- \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,{{Ly} + 1}}(k)}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k + {Ly} - p + 1} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{{Ly} + 1}(k)}}^{2}}}$ putting all partial derivatives at time k calculated by said u_(i)(k) with respect to the parameters to be tuned in the mathematical formula of said u_(i)(k) into the set {gradient of u_(i)(k)}; for said MIMO system, traversing all values of i in the positive integer interval [1, m] and obtaining the set {gradient of u₁(k)}, . . . , set {gradient of u_(m)(k)}, then putting them all into the set {gradient set}; said set {gradient set} is a set comprising all sets {{gradient of u₁ (k)}, . . . , {gradient of u_(m)(k) }}; said parameter self-tuning adopts neural network to calculate the parameters to be tuned in the mathematical formula of the control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T); the inputs of said neural network comprise at least one of: elements in said set {gradient set}, and elements in set {error set}; said set {error set} comprises at least one of: the error vector e(k)=[e₁(k), . . . , e_(n)(k)]^(T), and error function group of element e_(j)(k) (j=1, . . . , n) in said error vector e(k); said error function group of element e_(j) (k) comprises at least one of: the accumulation of the j-th error at time k and all previous times ${\sum\limits_{t = 0}^{n}{e_{j}(t)}},$ the first order backward difference of the j-th error e_(j)(k) at time k e_(j)(k)−e_(j)(k−1), the second order backward difference of the j-th error e_(j) (k) at time k e_(j)(k)−2e_(j)(k−1)+e_(j)(k−2), and high order backward difference of the j-th error e_(j)(k) at time k.
 3. The method as claimed in claim 2 wherein said neural network is BP neural network; said BP neural network adopts a single hidden layer structure, namely a three-layer network structure, comprising an input layer, a single hidden layer, and an output layer.
 4. The method as claimed in claim 2 wherein aiming at minimizing a system error function, said neural network adopts gradient descent method to update the hidden layer weight coefficients and the output layer weight coefficients, where the gradients are calculated by system error back propagation; independent variables of said system error function comprise at least one of: elements in the error vector e(k)=[e₁(k), . . . , e_(n)(k)]^(T), n desired system outputs, and n actual system outputs.
 5. The method as claimed in claim 4 wherein said system error function is defined as ${{\sum\limits_{{jy} = 1}^{n}{a_{jy}{e_{jy}^{2}(k)}}} + {\sum\limits_{{iu} = 1}^{m}{b_{iu}\Delta{u_{\iota u}^{2}(k)}}}},$ where e_(jy)(k) is the jy-th error at time k, Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), u_(iu)(k) is the iu-th control input at time k, a_(jy) and b_(iu) are two constants greater than or equal to zero, jy and iu are two positive integers.
 6. The method as claimed in claim 1 wherein said j-th error e_(j)(k) at time k is calculated by the j-th error function; independent variables of said j-th error function comprise the j-th desired system output and the j-th actual system output.
 7. The method as claimed in claim 6 wherein said j-th error function adopts at least one of: e_(j)(k)=y*_(j)(k)−y_(j)(k), e_(j)(k)=y*_(j)(k+1)−y_(j)(k), e_(j)(k)=y_(j)(k)−y*_(j)(k), and e_(j)(k)=y_(j)(k)−y*_(j)(k+1), where y*_(j)(k) is the j-th desired system output at time k, y*_(j)(k+1) is the j-th desired system output at time k+1, and y_(j)(k) is the j-th actual system output at time k. 